1. Circles and Arcs
Skill 46

2. Objective
HSG-C.1/2/5: Students are responsible for finding the measures of central angles, arcs, circumference, and arc length.

3. Definitions
In a plane, a circle is the set of all points equidistant from a given point called the center.
A diameter is a segment that contains the center of a circle and has both endpoints on the circle.
A radius is a segment that has one endpoint at the center and the other endpoint on the circle.
Congruent Circles have congruent radii.

4. Definitions
A central angle is an angle whose vertex is the center of the circle.
One type of arc is a semicircle which is half of a circle.
A minor arc is an arc that is smaller than a semicircle.
A major arc an arc that is larger than a semicircle.
Adjacent arcs are arcs of the same circle that have exactly one point in common.

5. Definitions
The circumference of a circle is the distance around the circle.
Coplaner circles that have the same center are concentric circles.
The measure of an arc is in degrees while arc length is a fraction of a circle’s circumference.
Congruent arcs are arcs that have the same measure and are in the same circle or in congruent circles.

6. Postulate 16: Arc Addition Postulate
The measure of the arc formed by two adjacent arcs is the sum of the measures of the two arcs.
𝒎 𝑨𝑩𝑪 =𝒎 𝑨𝑩 +𝒎 𝑩𝑪
Theorem 75: Circumference of a Circle
The circumference of a circle is π times the diameter.
𝑪=𝝅𝒅 or 𝑪=𝟐𝝅𝒓
Theorem 76: Arc Length of a Circle
The length of an arc of a circle is the product of the ratio measure of the arc to 360 and the circumference of the circle.
𝒍𝒆𝒏𝒈𝒕𝒉 𝑨𝑩 = 𝒎 𝑨𝑩 𝟑𝟔𝟎 ∗𝟐𝝅𝒓

7. 𝑨𝑫 , 𝑪𝑬 , 𝑨𝑪 , 𝑫𝑬 𝑨𝑪𝑬 , 𝑪𝑬𝑫 , 𝑬𝑫𝑨 , 𝑫𝑨𝑪 𝑨𝑪𝑫 , 𝑪𝑬𝑨 , 𝑬𝑫𝑪 , 𝑫𝑨𝑬
Example 1; Naming Arcs
a) What are the minor arcs of ⊙𝑂? O A C E D
𝑨𝑫 , 𝑪𝑬 , 𝑨𝑪 , 𝑫𝑬
b) What are the semicircles of ⊙𝑂?
𝑨𝑪𝑬 , 𝑪𝑬𝑫 , 𝑬𝑫𝑨 , 𝑫𝑨𝑪
c) What are the major arcs of ⊙𝑂?
𝑨𝑪𝑫 , 𝑪𝑬𝑨 , 𝑬𝑫𝑪 , 𝑫𝑨𝑬

8. Example 2; Finding Measures of Arcs
What is the measure of each arc in ⊙𝑂?
a) 𝐵𝐶 O C A D B
𝒎 𝑩𝑪 =𝒎∠𝑩𝑶𝑪
58ᵒ
𝒎 𝑩𝑪 =𝟑𝟐°
b) 𝐵𝐷
𝒎 𝑩𝑫 =𝒎 𝑩𝑪 +𝒎 𝑪𝑫
32ᵒ
𝒎 𝑩𝑫 =𝟑𝟐+𝟓𝟖
𝒎 𝑩𝑫 =𝟗𝟎°
c) 𝐴𝐵𝐶
𝑨𝑩𝑪 𝒊𝒔 𝒂 𝒔𝒆𝒎𝒊𝒄𝒊𝒓𝒄𝒍𝒆
𝑨𝑩𝑪 =𝟏𝟖𝟎°
d) 𝐴𝐵
𝒎 𝑨𝑩 =𝒎 𝑨𝑩𝑪 −𝒎 𝑩𝑪
𝒎 𝑨𝑩 =𝟏𝟖𝟎−𝟑𝟐
𝒎 𝑨𝑩 =𝟏𝟒𝟖°

9. Example 3; Finding Distance
A 2-ft-wide circular track for a camera dolly is set up for a movie scene. The two rails of the track form concentric circles the radius of the inner circle is 8 ft. How much farther does a wheel on the outer rail travel than a wheel on the inner rail of the track in one turn?
𝑪 𝒊𝒏𝒏𝒆𝒓 =𝟐𝝅𝒓
𝑪 𝒐𝒖𝒕𝒆𝒓 =𝟐𝝅𝒓
𝑪 𝒊𝒏𝒏𝒆𝒓 =𝟐 𝟖 𝝅
𝑪 𝒐𝒖𝒕𝒆𝒓 =𝟐 𝟏𝟎 𝝅
𝑪 𝒊𝒏𝒏𝒆𝒓 =𝟏𝟔𝝅
𝑪 𝒐𝒖𝒕𝒆𝒓 =𝟐𝟎𝝅
𝑫𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆=𝟐𝟎𝝅−𝟏𝟔𝝅
𝑫𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆=𝟒𝝅≈𝟏𝟐.𝟓𝟕
A wheel on the outer track travels ft. farther than a wheel on the inner track.

10. Example 4; Finding Arc Length
What is the length of each arc bold arc? Leave your answer in terms of pi.
a) Diameter = 16 in. b) Radius = 15 cm. O Y X O X Y P
𝒍𝒆𝒏𝒈𝒕𝒉 𝑿𝒀 = 𝒎 𝑿𝒀 𝟑𝟔𝟎 ∗𝝅𝒅
𝟐𝟒𝟎°
𝒍𝒆𝒏𝒈𝒕𝒉 𝑿𝑷𝒀 = 𝒎 𝑿𝑷𝒀 𝟑𝟔𝟎 ∗𝟐𝝅𝐫
𝒍𝒆𝒏𝒈𝒕𝒉 𝑿𝒀 = 𝟗𝟎 𝟑𝟔𝟎 ∗𝟏𝟔𝝅
𝒍𝒆𝒏𝒈𝒕𝒉 𝑿𝑷𝒀 = 𝟐𝟒𝟎 𝟑𝟔𝟎 ∗𝟐 𝟏𝟓 𝝅
𝒍𝒆𝒏𝒈𝒕𝒉 𝑿𝒀 =𝟒𝝅 𝒊𝒏.
𝒍𝒆𝒏𝒈𝒕𝒉 𝑿𝑷𝒀 =𝟐𝟎𝝅 𝒄𝒎

11. #46: Circles and Arcs
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